Level 3 Mathematics and Statistics (Statistics) (91585) · PDF fileLevel 3 Mathematics and Statistics (Statistics), 2013 91585 Apply probability concepts in solving problems ... With

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Level 3 Mathematics and Statistics (Statistics), 2013

91585 Apply probability concepts in solving problems

9.30 am Wednesday 20 November 2013 Credits: Four

Achievement Achievement with Merit Achievement with ExcellenceApply probability concepts in solving problems.

Apply probability concepts, using relational thinking, in solving problems.

Apply probability concepts, using extended abstract thinking, in solving problems.

Check that the National Student Number (NSN) on your admission slip is the same as the number at the top of this page.

You should attempt ALL the questions in this booklet.

Show ALL working.

Make sure that you have the Formulae and Tables Booklet L3–STATF.

If you need more space for any answer, use the page(s) provided at the back of this booklet and clearly number the question.

Check that this booklet has pages 2 – 10 in the correct order and that none of these pages is blank.

YOU MUST HAND THIS BOOKLET TO THE SUPERVISOR AT THE END OF THE EXAMINATION.

You are advised to spend 60 minutes answering the questions in this booklet.

QUESTION ONE: SPORTS

(a) A local school has decided to hold a sports week. Two different year levels of students (Years 9 and 13) were surveyed before the sports week, and students were asked if they played any sports:

• 52%ofthestudentssurveyedwereYear9students

• 84%oftheYear9studentssurveyedsaidthattheyplayedatleastonesport

• 62%oftheYear13studentssurveyedsaidthattheyplayedatleastonesport.

(i) What percentage of the students surveyed said that they played at least one sport?

(ii) If a student randomly selected from those surveyed said that they played no sports, are they more likely to be a Year 9 or a Year 13 student?

Support your answer with appropriate statistical statements.

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Mathematics and Statistics (Statistics) 91585, 2013


(b) The three most popular sports played at the school are netball, tennis and kilikiti.

Ofthe195studentsattheschool:

• 45studentsdonotplaynetball,tennis,orkilikiti

• 5studentsplaynetball,tennis,andkilikiti

• 8studentsplaytennisonly

• 20studentsplaybothtennisandnetball,andmayalsoplaykilikiti

• 12studentsplaybothtennisandkilikiti,andmayalsoplaynetball

• 35studentsplaybothnetballandkilikiti,andmayalsoplaytennis

• 50studentsplaykilikiti.

(i) Calculate the percentage of students at the school who play tennis.

(ii) If two different students from the school are selected at random, without replacement, calculate the probability that they both play netball.

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QUESTION TWO: INJURIES

(a) Theresultsofthe2007/08ActiveNZSurvey,anddatacollectedbytheAccidentCompensationCorporation(ACC)duringtheperiod1July2007to30June2008,havebeenused to create the following table:

Estimates for New Zealand adults Tennis NetballNumberwhoplayedthissport 311662 123994Numberofplayersinjuredwhileplaying this sport 7354 15143

(i) IsaNewZealandadultmorelikelytobeinjuredwhileplayingtennisorwhileplayingnetball?

Support your answer with appropriate statistical statements.

(ii) With regards to probability theory, explain why it is not possible to use this informationtocalculatetheprobabilityofaNewZealandadultbeinginjuredwhileplaying tennis OR netball.

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(b) Duringsportsweekatalocalschool,theschoolnurserecordedinformationaboutinjuries.

(i) LetAbetheevent“astudentisinjuredplayingtennis”. LetBbetheevent“astudentisinjuredplayingnetball”.

From the information the school nurse has recorded, it can be deduced that:

P(A ∪ B) = 0.35 and P(A ∪ B' ) = 0.90.

Determinetheproportionofstudentswhowereinjuredplayingtennisduringsportsweek.

(ii) The nurse’s records also show that:

• 12%oftheinjuriesobtainedwereserious

• ofthestudentswhowereseriouslyinjured,52%wereinjuredwhileplayingrugby

• studentsweretwiceaslikelytobehavebeenplayingrugbyiftheywereseriouslyinjuredthaniftheywerenotseriouslyinjured.

Calculatetheprobabilityofastudentbeinginjuredwhileplayingrugbyduringsports

week.

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This page has been deliberately left blank.

The examination continues on the following page.

QUESTION THREE: GAMES

(a) Players are strongly advised to warm up before playing sports games to reduce their risk of injuryfromplayingthegame.

Foraparticularsportsteamof20players:

• 14oftheplayerswarmedupbeforethelastgame

• 5oftheplayerswereinjuredduringthelastgame

• 2oftheplayersdidnotwarmupandwerenotinjuredduringthelastgame.

Using this information, calculate the probability that a randomly chosen player from the team wasinjured,giventhattheplayerdidnotwarmupbeforethelastgame.

Question Three continues on the following page.

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(b) At a local school, a board game is played during lunch break. Each turn consists of rolling a single six-sided die. The player moves as many squares as shown on the top face of the die. Tofinishthegame,aplayermustrolltheexactnumberrequiredtolandonthefinalsquare,numbered100.Ifthedieshowsanumbergreaterthantheremainingnumberofsquares,theplayercannotmove,andmustwaituntiltheirnextturntotryandfinishthegame.

Forexample,ifaplayerisonsquare97,theycouldfinishthegameinonerollbyrollinga3.Theycouldfinishthegameintworolls,eitherbyrollinga2onthefirstturn,andthenrollinga1onthesecondturn,orbyrollinga5onthefirstturn(whichmeanstheycannotmoveforthat turn), and then rolling a 3 on the second turn.

(i) A student designed a computer simulation to investigate the distribution of the number ofturnsneededtofinishthegamefromsquare 95.

150trialsarecarriedout.Theresultsaregraphedbelow.

10

Fre

quen

cy

Number of rolls required

5

10

15

20

25

30

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Rolls required to finish the game from square 95

Usetheseresultstoestimatetheprobabilityoffinishingthegamefromsquare 95in5or fewer rolls.

(ii) Explainwhythetheoreticalprobabilityoffinishingthegamefromsquare 95 in exactly

two rolls is 536

.

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(iii) Anotherstudentattheschoolthinksthattheformulabelowcanbeusedtofindthetheoreticalprobabilityoffinishingthegamefromsquare 95 in r number of rolls.

P(R = r) = 5

6⎛⎝⎜

⎞⎠⎟

r−116

⎛⎝⎜

⎞⎠⎟

Discuss whether this student is correct in her thinking.

Youmaywishtoincludediagramsand/orcalculationsaspartofyourdiscussion.

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QUESTION NUMBER

Extra paper if required.Write the question number(s) if applicable.

91

58

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Level 3 Mathematics and Statistics (Statistics) (91585) · PDF fileLevel 3 Mathematics and Statistics (Statistics), 2013 91585 Apply probability concepts in solving problems ... With